On the geometry of the anti-canonical bundle of the Bott-Samelson-Demazure-Hansen varieties

Let $G$ be a semi-simple simply connected algebraic group over the field $\mathbb{C}$ of complex numbers. Let $T$ be a maximal torus of $G,$ and let $W$ be the Weyl group of $G$ with respect to $T$. Let $Z(w,\underset{‾}{i})$ be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple $\underset{‾}{i}$ associated to a reduced expression of an element $w\in W\mathrm{.}$ We prove that for the tuple $\underset{‾}{i}$ associated to any reduced expression of a minuscule Weyl group element $w,$ the anti-canonical line bundle on $Z(w,\underset{‾}{i})$ is globally generated. As consequence, we prove that $Z(w,\underset{‾}{i})$ is weak Fano. Assume that $G$ is a simple algebraic group whose type is different from $A_2\mathrm{.}$ Let $S=\{{\alpha }_1,\cdots ,{\alpha }_n\}$ be the set of simple roots. Let $w$ be such that support of $w$ is equal to $S\mathrm{.}$ We prove that $Z(w,\underset{‾}{i})$ is Fano for the tuple $\underset{‾}{i}$ associated to any reduced expression of $w$ if and only if $w$ is a Coxeter element and $w^{-1}({\sum }_{t=1}^n{\alpha }_t)\in -S$.